The exponential expression [latex]b^{n}[/latex], where the exponent [latex]n[/latex] is a positive integer, is used to represent the repeated product of [latex]n[/latex] factors of the base [latex]b[/latex].  For example

[latex]5^{3}=5 \cdot 5 \cdot 5 = 125[/latex].

If the exponent is not a positive integer, we can use the following properties of exponents to evaluate the expression.

For [latex]b \neq 0[/latex],
[latex]b^{-n}= \frac{1}{b^{n}}[/latex]
[latex]b^{0}=1[/latex]

A shorthand method of writing very small and very large numbers is called scientific notation, in which we express numbers in terms of exponents of 10. A number is said to be in scientific notation if it is written in the form

[latex]a \times 10^{n}[/latex] where [latex]1 \leq |a| < 10[/latex].

The number [latex]a[/latex] can be positive or negative, but will always have exactly one nonzero digit to the left of the decimal point. For example, [latex]3.2 \times 10^{3}[/latex] and [latex]9.01 \times 10^{-3}[/latex] are in scientific notation. To rewrite these numbers in standard notation, we can evaluate the powers of [latex]10[/latex] and perform the multiplication.

[latex]3.2 \times 10^{3} = 3.2 \times 1000 = 3,200[/latex]

[latex]9.01 \times 10^{-3} = 9.01 \times \frac{1}{10^{3}} = \frac{9.01}{1000} = 0.00901[/latex]

In both of these examples, the decimal point moved [latex]3[/latex] places. In converting [latex]3.2 \times 10^{3}[/latex] to [latex]3,200[/latex], we multiply by [latex]10[/latex] three times, moving the decimal point [latex]3[/latex] places to the right.  In converting [latex]9.01 \times 10^{-3}[/latex] to [latex]0.00901[/latex], we divided by [latex]10[/latex] three times, moving the decimal point [latex]3[/latex] places to the left.

Converting from Scientific to Standard Notation

To convert a number in scientific notation to standard notation, move the decimal point n places to the right if [latex]n[/latex] is positive, or [latex]|n|[/latex] places to the left if [latex]n[/latex] is negative. Add zeros as needed.

Converting from Standard to Scientific Notation

To write a number in scientific notation, we reverse this process. To begin, move the decimal point to the right of the first nonzero digit in the number. Write the digits as a decimal number between 1 and 10. Count the number of places [latex]n[/latex] that you moved the decimal point. Multiply the decimal number by 10 raised to a power of [latex]n[/latex]. If you moved the decimal left as in a very large number, [latex]n[/latex] is positive. If you moved the decimal right as in a small large number, [latex]n[/latex] is negative.

For example, consider the number 2,780,418. Move the decimal left until it is to the right of the first nonzero digit, which is 2.

We obtain 2.780418 by moving the decimal point 6 places to the left. Therefore, the exponent of 10 is 6, and it is positive because we moved the decimal point to the left. This is what we should expect for a large number.

Working with small numbers is similar. Take, for example, the radius of an electron, 0.00000000000047 m. Perform the same series of steps as above, except move the decimal point to the right.

Be careful not to include the leading 0 in your count. We move the decimal point 13 places to the right, so the exponent of 10 is 13. The exponent is negative because we moved the decimal point to the right. This is what we should expect for a small number.

Watch the following video to see more examples of writing numbers in scientific notation.

Calculators sometimes represent values in scientific notation in a slightly different format. Try multiplying

[latex]500,000 \times 600,000[/latex]

On some calculators your answer may read 3E11.  This means the same thing as

[latex]3 \times 10^{11} = 300,000,000,000[/latex].